Heat transport in ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CuO}}_{6+\delta }$: Departure from the Wiedemann-Franz law in the vicinity of the metal-insulator transition

We present a study of heat transport in the cuprate superconductor ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CuO}}_{6+\delta }$ at sub-Kelvin temperatures and in magnetic fields as high as 25 T. In several samples with different doping levels close to optimal, the linear-temperature term of thermal conductivity was measured both at zero field and in presence of a magnetic field strong enough to quench superconductivity. The zero-field data yields a superconducting gap of reasonable magnitude displaying a doping dependence similar to the one reported in other families of cuprate. The normal-state data together with the results of the resistivity measurements allows us to test the Wiedemann-Franz (WF) law, the validity of which was confirmed in an overdoped sample in agreement with previous studies. In contrast, a systematic deviation from the WF law was resolved for samples displaying either a lower doping content or a higher disorder. Thus, in the vicinity of the metal-insulator crossover, heat conduction in the zero-temperature limit appears to become significantly larger than predicted by the WF law. Possible origins of this observation are discussed.

Figure 1

Thermal conductance of a gold wire for $H=0$ (open circle) and for $H=25\mathrm{T}$ (solid squares) compared to the magnitude expected by the WF law (solid lines).

Figure 2

Upper panel: Thermal conductivity of sample UD3 measured a first time (solid triangles) and a second time using another setup three months later (open squares). Lower panel: Temperature dependence of the resistivity of the same sample measured under the same conditions. Solid (dotted) line is the first (second) measurement.

Figure 3

(a) Temperature dependence of the resistivity at zero magnetic field. (b) Temperature dependence of the resistivity at different magnetic fields for sample UD3. The inset shows a magnetoresistivity curve obtained around 0.3 K (c) The temperature dependence of resistivity for the four samples studied at $B=25\mathrm{T}$. Note that the resistivity is almost flat as a function of temperature for all the samples investigated.

Figure 4

(a) Thermal conductivity at low temperature for different samples plotted as $\kappa ∕T$ versus $T^2$. The dotted lines represent linear extrapolations to $T=0$. (b) Doping dependence of the linear term of the thermal conductivity extrapolated to $T=0$ (squares).

Figure 5

Comparison of thermal conductivity in the superconducting state (open symbols) and in the normal state (solid symbols) for (a) an overdoped sample, (b) an optimally doped sample, and (c) and (d) two slightly underdoped samples. Note the increase of $\kappa \left(H\right)$ seen in the overdoped sample and the decrease of $\kappa \left(H\right)$ in the underdoped samples. Arrows mark the expected value for $\kappa ∕T$ in the normal state according to the WF law. The zero-temperature extrapolation becomes significantly larger than this value in the underdoped samples (see text).

Figure 6

Lorentz number normalized by the Sommerfeld value $L_0$ versus $T^2$ for different samples at various doping level in the normal state. Dashed line is the WF expectation at $T=0$.

Figure 7

(a) The main panel shows the doping dependence of the Lorentz ratio normalized by $L_0$ for all Bi-2201 samples (full squares). The inset is a comparison with data reported in previous studies for PCCO (Ref. 19), Tl-2201 (Ref. 21), and LSCO (Ref. 22). The horizontal line is the WF expectation. The shaded area marks the crossover from metal-to-insulator in La-doped Bi-2201 (Ref. 43). (b) Lorentz ratio normalized by $L_0$ versus residual resistivity ratio for Bi-2201 (see text).